Almost-crystallographic subgroups of Bn/Γ3(Pn) and infra-nilmanifolds with cyclic holonomy

Abstract

Let n≥3. In this work we study almost-crystallographic subgroups H˜n,3=σ−1(H)/Γ3(Pn) of Bn/Γ3(Pn) with cyclic holonomy group H, where Bn is the Artin braid group, σ:Bn→Sn is the natural projection and Γ3(Pn) is the third-step element of the lower central series of the Artin pure braid group Pn. We study in detail the holonomy representation of H˜n,3 describing it as a matrix representation. As an application, when H is a cyclic group of order a power of 2, we determine whether the infra-nilmanifold X with fundamental group H˜n,3 is orientable and if it admits spin structure.